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International Journal of Computer Applications (0975 – 8887)
Volume 89 – No 18, March 2014
On Generalized d-Closed Sets
H. Jude Immaculate
I. Arockiarani
Research Scholar
Nirmala College for Women
Coimbatore.
Department of Mathematics
Nirmala College for Women
Coimbatore.
ABSTRACT
In this paper we present a new class of sets and functions
namely gd-closed sets, gd-irresolute functions in the light of
d-open sets in topological spaces. Further some of their
characterizations are investigated with counter examples.
Keywords
gd-closed
functions.
sets,
gd-continuous
functions,
Definition 1.2: A subset A of a topological space (X,
called
A generalized closed set[12] (briefly g-closed) set if
cl(A)  U whenever A  U and U is open.
2.
A generalized semiclosed set[3] (briefly gs-closed if
𝑠cl(A)  U whenever A  U and U is open.
3.
A -generalized closed set [13](briefly  g-closed) if
 cl(A)  U whenever A  U and U is open.
4.
A generalized preclosed set [14](briefly gp-closed) if
pcl(A)  U whenever A  U and U is open.
5.
A generalized pre regular closed set[9] (briefly
gpr-closed) if pcl(A)  U whenever A  U and U is
regular open
6.
A -regular generalized closed set[12] (briefly
 gr –closed) if  cl(A)  U whenever A  U and U
is regular open
7.
A regular generalized closed set [18](briefly rg-closed)
if cl(A)  U whenever A  U and U is regular-open.
1. INTRODUCTION
Preliminaries: We present here relevant preliminaries
required for the progress of this paper
Definition 1.1: A subset A of a topological space (X,  ) is
called
1. Preclosed set[15] if cl(int(A))  A, preopen set if
A  int(cl(A))
2.

-open set [17] if A  int(cl(int(A))),
set if cl(int(cl(A)))  A.

4. Semiopen set [11] if A
if int(cl(A)  A.
 cl(int(A)), semiclosed set
5. Semi-pre-open set[1] if A 
semi-pre-closed set if int(cl(int(A)))
2. GENERALIZED D-CLOSED SETS
Definition 2.1: A subset A of a space X is called Generalized
d-closed set if dcl(A)  U whenever A  U and U is open.
The class of all generalized d-closed sets is denoted by
GDC(X).
Definition 2.2:
i.
The gd-closure of a subset A of X is denoted by
gdcl(A) is the smallest gd-closed set containing A.
ii.
The gd-interior of a subset A of X is denoted by
gdint(A) is the largest gd-open set contained in A.
-closed
3. Regular open set[19] if A=int(cl(A)), regular closed
set if A=cl(int(A)))
) is
1.
gd-irresolute
Many different forms of continuous functions have been
introduced over years. Most of them involve the concept of
g-closed sets, -open sets, semiopen sets, sg-open sets, etc. In
1987, Bhattacharya and Lahiri [5] introduced the class of
semi-generalized closed sets. In 1990, Arya and
Nour[3]defined generalized semiclosed sets. The concept of
generalized closed sets was first initiated by Levine in
1970[12]. The notion of b-open sets was defined by
D. Andrijevic in 1996[2].In this paper a new class of sets
called gd-closed sets has been introduced using the concept of
d-closed sets by I.Arockiarani et al[10]. Further we study the
basic properties of gd-closed sets. Using this new concept of
sets we have introduced new class of functions called
gd-continuous and gd-irresolute functions. Some of its basic
properties and composition of functions is also discussed here.

Proposition 2.3[9]: The intersection of a open set and d-open
set is d-open set.
Proposition 2.4:
cl(int(cl(A))),
 A.
6. d-open set[10] if A  scl(int(A))  sint(cl(A))
i.
Every closed set is gd-closed set.
ii.
Every d-closed set is gd-closed set.
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International Journal of Computer Applications (0975 – 8887)
Volume 89 – No 18, March 2014
Example 2.15: Let X= {a, b, c},  = {X,
Proof:
i.
ii.
Let A be closed set such that A  U where U is
open in (X,  ). Since A is closed cl(A)=A  U.
But every closed set is d-closed. Hence A is
gd-closed.
Let A be d-closed set such that A  U where U is
open in (X,  ). Since A is d-closed dcl(A)=A  U.
Hence A is gd-closed. The converse of the above
result need not be true may seen by the following
example.
Example 2.5: Let X= {a, b, c},  = {X,  , {a}, {a, b}}.
A= {a, c} is gd-closed but not d-closed or closed.
Proposition 2.6: Every  -closed set is gd-closed set.
Proof: Let A be  -closed set such that A  U where U is
open in (X,  ). Since A is  -closed  cl(A)=A  U. But
every -closed set is d-closed. Hence A is gd-closed. The
converse of the above result need not be true may seen by the
following example.
Example 2.7: Let X= {a, b, c},  = {X,
A= {b} is gd-closed but not  -closed.
 , {a}, {b, c}}.

 , {b, c}}. A= {b}
g-closed set.
Proposition 2.16: Every gp-closed set is gd-closed set.
Proof: Let A be a gp-closed set then pcl(A)  U whenever A
 U and U is open in (X,  ). Since every pre closed set is
d-closed set we have dcl(A)  pcl(A)  U. Hence A is
gd-closed. The converse of the above result need not be true
may seen by the following example.
Example2.17: Let X= {a, b, c},  = {X,  , {a}, {b}, {a,
b}}. A= {a} is gd-closed set but not gp-closed set.
Proposition2.18: Every gpr-closed set is gd-closed set.
Proof: Let A be a gpr-closed set then pcl(A)  U whenever
A  U and U is regular open in (X,  ). Every regular open is
open hence A  U and U is open. Since every pre closed set
is d-closed set we have dcl(A)  pcl(A)  U. Hence A is
gd-closed. The converse of the above result need not be true
may seen by the following example.
Example 2.19: Let X= {a, b, c},  = {X,  , {a}, {b}, {a,
b}}. A= {a} is gd-closed set but not gpr-closed set.
Proposition 2.20: Every  gr-closed set is gd-closed set.
Proposition 2.8: Every preclosed set is gd-closed set.
Proof: Let A be preclosed set such that A  U where U is
open in (X,  ). Since A is preclosed set pcl(A)=A  U. But
every preclosed set is d-closed set. Hence A is gd-closed. The
converse of the above result need not be true may seen by the
following example.
Example 2.9: Let X= {a, b, c},  = {X,  , {a}, {a, b}}.
A= {a, c} is gd-closed set but not preclosed set.
Proposition 2.10: Every semiclosed set is gd-closed set.
Proof: Let A be a semiclosed set such that A  U where U is
open in (X,  ). Since A is semiclosed set scl(A)=A  U.
But every semiclosed set is d-closed set. Hence A is
gd-closed. The converse of the above result need not be true
may seen by the following example.
Example 2.11: Let X= {a, b, c},  = {X,  , {a}, {b, c }}.
A= {b} is gd-closed set but not semiclosed set.
Proposition 2.12: Every g-closed set is gd-closed set.
Proof: Let A be a g-closed set then cl(A)  U whenever A
 U and U is open in (X,  ). Since every closed set is
d-closed set we have dcl(A)  cl(A)  U. Hence A is
gd-closed. The converse of the above result need not be true
may seen by the following example.
Example 2.13: Let X= {a, b, c},  = {X,
A= {b} is gd-closed set but not g-closed set.
is gd-closed set but not
 , {a}, {a, b}}.
Proposition 2.14: Every  g-closed set is gd-closed set.
Proof: Let A be a  gr-closed set then  cl(A)  U
whenever A  U and U is regular open in (X,  ). Every
regular open is open hence A  U and U is open. Since every
 -closed set is d-closed set we have dcl(A)  cl(A) U.
Hence A is gd-closed. The converse of the above result need
not be true may seen by the following example.
Example2.21: Let X= {a, b, c},  = {X,  , {a}, {b}, {a,
b}}. A= {a} is gd-closed set but not  gr-closed set.
Proposition 2.22: Every rg-closed set is gd-closed set.
Proof: Let A be a rg-closed set then cl(A)  U whenever A
 U and U is regular open in (X,  ). Every regular open is
open hence A  U and U is open. Since every closed set is
d-closed set we have dcl(A)  cl(A)  U. Hence A is
gd-closed. The converse of the above result need not be true
may seen by the following example.
Example 2.23: Let X= {a, b, c},  = {X,  , {a}, {b}, {a,
b}}. A= {a} is gd-closed set but not rg-closed set.
Proposition 2.24: Every gs-closed set is gd-closed set.
Proof: Let A be a gs-closed set then scl(A)  U whenever A
 U and U is open in (X,  ). Since every semiclosed set is
d-closed set we have dcl(A)  scl(A)  U. Hence A is
gd-closed. The converse of the above result need not be true
may seen by the following example.
Example 2.25: Let X= {a, b, c},  = {X,
is gd-closed set but not gs-closed set.
 , {b, c}}. A= {b}
Proof: Let A be a g-closed set then  cl(A)  U whenever
A  U and U is open in (X, ). Since every  -closed set is dclosed set we have dcl(A)   cl(A)  U. Hence A is dclosed. The converse of the above result need not be true may
seen by the following example.
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International Journal of Computer Applications (0975 – 8887)
Volume 89 – No 18, March 2014
From the above examples we have the following diagram
1
2
12
3
11
4
13
5
10
9
6
8
7
Figure. 1 Relation between generalized d-closed set and other existing closed sets
1.gs closed set 2.g closed set 3.  gr closed set 4. rg closed set 5. gpr closed set 6. gp closed set
7.  g-closed set 8.  -closed set 9.semiclosed set 10.d-closed set 11.preclosed set 12.closed set
13.gd-closed set.
Lemma 2.26: For a subset A of a space X the following hold
b.
 scl(int(A))
dint(A)=A  (scl(int(A)  sint(cl(A))
c.
dcl(X-A)=X-dint(A)
a.
dcl(A)=A  (sint(cl(A)
Theorem2.27: A set A is gd-open if and only if F  dint(A)
whenever F is closed and F  A.
Proof: Let A be gd-open and suppose that F  A and F is
closed. Then X-A is gd-closed set. Thus gdcl(X-A)  X-F .
But dcl(X-A)=X-dint(A). Thus X-dint(A)  X-F. Hence F 
dint(A).Conversely, let F be closed set with F  dint(A)
whenever F  A. F  A(X-A)  (X-F). Also F 
dint(A). Thus (X-dint(A))=dcl(X-A)  X-F where (X-F) is
open. Thus A is gd-open.
Theorem 2.28: Let A be closed subset of (X,  ). Then
dcl(A)-A does not contain any non empty closed set.
Proof: Let A be generalized d-closed and F be a non empty
closed set such that F  dcl(A)-A. Then F  dcl(A)  F
dcl(A) and F 
c
A c . F  A  F is open. Then dcl(A)
c
 F c F  (dcl ( A)) .F  dcl(A)  (dcl ( A)) c F
  which is a contradiction. Hence dcl(A)-A contains no
c
non empty closed set.
Theorem 2.29: Let A be a gd-closed subset of (X, ) and A
 B  dcl(A), then B is gd-closed.
Proof: Let A  B  dcl(A) then dcl(A)=dcl(B).Since A is
gd-closed, dcl(A)  G where A  G; G is open in X. Let B
 G and G is open in X, since A is gd-closed and since
dcl(A)=dcl(B), dcl(B)  G. Thus B is gd-closed.
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International Journal of Computer Applications (0975 – 8887)
Volume 89 – No 18, March 2014
Theorem 2.30: A gd-closed set A is d-closed if and only if
dcl(A)-A is closed
Proof: Let A be gd-closed. If A is d-closed then dcl(A)-A= 
is a closed set. Thus dcl(A)-A is closed. Conversely, let
dcl(A)-A be closed. .dcl(A)-A is closed subset of itself hence
dcl(A)-A contain any non empty closed subset thus dcl(A)-A
=  (i.e) A=dcl(A).Hence A is d-closed.
Definition 2.31: Let B  A  X, then we say B is gd-closed
relative to A if
in A
dcl A (B)  U when B  U and U is open
Definition 3.3: A function f: (X,  )  (Y,  ) be
d-irresolute if the pre image of every d-open set in Y is
d-open in X.
Definition 3.4: A function f: (X,  )  (Y,  ) is called
d -continuous if the inverse image of every closed set in Y is
d-closed in X
Theorem 3.5: Let f:(X,  )  (Y,  ) be d-continuous,
then f is gd-continuous but not conversely.
Proof: Let V be a open set in Y. Since f is d-continuous then
f
1
(V ) is d-open in X. But every d-open set is gd-open in
1
Theorem 2.32: Let B  A  X where A is gd-closed and
open. Then B is gd-closed relative to A if and only if B is
gd-closed.
X hence f
continuous.
Proof: Let B  A  X, where A is gd-closed and open and
let B is gd-closed relative to A. Since B  A and A is gdclosed and open dcl(A)  A. Thus dcl(B)  dcl(A)  A.
Y= {a, b, c},  = {{X,  , {a}, {b}, {a, b}}. Let f:(X,  )
(Y,  ) be identity map. f is gd-continuous but not d-
dcl A (B) =A  dcl(B) so dcl(B)= dcl A (B)  A. If
B is gd-closed relative to A and U is a open subset of X such
that B  U then B=B  A  U  A where U  A is open in
A. Thus B is gd-closed relative to A. dcl(B)= dcl A (B)
 U  A  U. Thus B is gd-closed in X. Conversely, Let B
is gd-closed in X and U is open subset of A such that B  U.
Let U=V  A for some open subset V of X.As B  V & B is
gd-closed in X ,dcl(B)  V. Thus dcl A (B) =dcl(B)  A
 V  A=U. Hence B is gd-closed relative to A.
Corollary 2.33: Let A be a gd-closed set which is also open
then A  F is gd-closed whenever F is d-closed.
Proof: Let A be a gd-closed and open then dcl(A)  A. But
A  dcl(A) thus A=dcl(A).Hence A  F is d-closed. Every
d-closed is gd-closed. A  F is gd-closed.
Remark 2.34: Intersection of two gd-closed sets need not be
gd-closed.
Example 2.35: Let X={a, b, c},  ={X,  ,{a}}. A= {a, b}
which is gd-closed set. B= {a, c} which is gd-closed set. But
A  B= {a} is not gd-closed.
Remark 2.36: Union of two gd-closed sets need not be
gd-closed.
(V )
is gd-open in X for VY. Thus f is
Example 3.6: Let X= {a, b, c},
continuous because
f
1
(b)={b}
Theorem 3.7: Let f: (X,

)

= {X,
 , {a}, {a, b}}.
DO(X).

(Y,

) be a continuous

function. If f is gd-continuous then for each point x X and
for each open set V in Y containing f(x), there exists a
gd-open set U containing x such that f(U)  V
The proof follows immediately.
Theorem 3.8: If the bijective map f :(X,  )  (Y,
d-irresolute and open, then f is gd-irresolute.
) is
1
V is sgd-closed in Y dcl(A)  f(U).Therefore f (dcl(V)
 U and since f is d-irresolute and dcl(V) is d-closed in Y,
f
1
(dcl(V) is d-closed set in X. Thus dcl(
1
1
f
1
dcl( f (dcl(V))) = f (dcl(V)  U. Hence
gd-closed and thus f is gd irresolute.
Remark 3.9: The concept of
gd-irresoluteness are independent.
(V ) ) 
f
1
d-irresoluteness
(V ) is
and
Example 3.10: Let X= {a, b, c},  = {X,  , {a}, {a, b}}.
Y= {a, b, c},  = {{X,  , {a}}. Let f:(X,  )  (Y,  )
be identity map. Here f is d-irresolute but not gd-irresolute
 GDO(Y) but
f
1
GDO(X).
since V={c}
3. gd-Continuous and gd-Irresolute
Functions
Definition 3.1: A function f :(X,  )  (Y,  ) is
gd-irresolute but not d-irresolute, since V={b}
Definition 3.2: A function f: (X,  )  (Y,  ) is called
gd-irresolute if the pre image of every gd-closed set of Y is
gd-closed in X.

Proof: Let V be gd-closed in Y and let f 1 (V )  U where
U is open in X. Hence V  f(U) holds. Since f(U) is open and
Example 2.37: Let X={a, b, c},  ={X,  ,{a}, {c}, {a, c}},
A={a} which is gd-closed set. B={c} which is gd-closed set.
But A  B = {a, c} is not gd-closed.
called
gd-continuous if the pre image of every closed set of Y is
gd-closed in X.
gd-
Example 3.11: Let X= {a, b, c},

) (Y,

𝑓 −1 (𝑏)={a,
(c)={c}

= {X,
 , {a}}. Let f:(X,
) be defined by f(a)=f(c)=b, f(b)=c. Here f is
c}D
 C(X).
 DC(X) but
Theorem 3.12: Let f:(X,  )  (Y,  ) be a map from a
topological space (X,  ) into a topological space (Y,  ),
then the following are equivalent.
a)
b)
in X.
f is gd-continuous.
The inverse image of each open set in Y is gd-open
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International Journal of Computer Applications (0975 – 8887)
Volume 89 – No 18, March 2014
The proof follows immediately from the definition.
Theorem 3.13: Let f:(X,  )  (Y,
)(Z,  ) be any two functions then

) and g: (Y,

d- T 1 so f
1
(V ) is closed in X. Every closed is d-closed
2
hence f is d-irresolute.
1.
gof: (X,  )( Z,  ) is gd-continuous if g is
continuous and f is gd-continuous.
2.
gof: (X,  )  ( Z,  ) is gd-irresolute if g is
gd-irresolute and f is gd- irresolute.
1.
3.
gof: (X,  )  ( Z,  ) is gd-continuous if g is
gd-continuous and f is gd-irresolute.
gof: (X,  )  ( Z,  ) is gd-continuous if g is
gd-continuous and f is gd-continuous.
2.
gof: (X,  )  ( Z,  ) is d-continuous if g is
gd-continuous and f is d-continuous.
3.
gof: (X,  )  ( Z,  ) is gd-continuous if g is
gd-irresolute and f is gd-continuous.
4.
gof: (X,  )  ( Z,  ) is d-continuous if g is
gd-irresolute and f is d-continuous.
Proof:
1.
 ). g 1 (V )
Let V closed set in (Z,
2.
3.
be a d- T 1 space then the composition
2
is closed in

1
1
), since g is continuous. Thus f ( g (V ) )
is gd-closed in (X,) since f is gd-continuous. Thus
gof is gd-continuous.
(Y,
Theorem 3.20: Let X and Z be any topological spaces and Y
Let V be gd-closed set in (Z,  ), since g (V ) is
gd-closed in (Y,  ).Since f is gd-irresolute, is
gd-closed in (X,  ). Thus gof is gd-irresolute.
1
Proof:
1.
in Y. But Y is a d- T 1 space thus g 1 (U ) is closed
2
1
1
1
1
( g (V ) ) is gd-closed in (X,
gd-continuous.

). Thus gof is
2.
1
in Y. Since f is d-continuous f ( g 1 (V ) ) is
d-closed in X. Thus gof is d-continuous.
if every gd-closed set is closed.
Definition 3.15 [13]: A topological space X is called a T
1
2
3.
space if every g-closed sets is closed.
2
T
1
g (U ) is closed in Y.Since f is gd-continuous
Every d- T space is a T space
1
1
4.
2
Let U be gd-closed in Z, then g 1 (U ) is gd-closed
in Y. But Y is a d- T 1 space thus g 1 (U ) is closed
T
Every d- T space is a  b space.
1
2
1
2

1
( g 1 (V ) ) is gd-closed in X. Thus gof is gdcontinuous.
f
Proposition 3.18:
Example 3.18: Let X={a, b, c},
Let U be gd-closed set in Z, then g 1 (U ) is
gd-closed in Y. But Y is a d- T 1 space thus
Definition 3.16 [7]: A topological space is called a  b
space if every  g-closed set is closed.
2.
Let U be closed set in Z, then g 1 (U ) is gd closed
2
1
2
2
(V ) ) is
in Y. But Y is a d- T 1 space thus g 1 (U ) is closed
Definition 3.14: A topological space X is called a d- T space
1.
1
in Y. Since f is gd-continuous f ( g
gd-closed in X. Thus gof is gd-continuous.
Let V be closed set in (Z,). Thus g (V ) is
gd-closed in (Y,  ).Since f is gd-irresolute
f
Let U be closed set in Z, then g 1 (U ) is gd-closed
= {X,
,
{a} {b},{a,
T
b}}.Then the collection is T and  b but not d- T since
1
1
2
2
A={b} is gd-closed but not closed.
Theorem 3.19: Let f:(X,  )  (Y,  ) be gd-irresolute
function then f is d-irresolute function if X is d- T space.
1
2
Proof: Let V be d-closed set in Y. Since V is gd-closed in Y
and f is gd-irresolute, f 1 (V ) is gd-closed in x. But X is
in Y. Since f is d-continuous f ( g 1 (V ) ) is
d-closed in X. Thus gof is d-continuous in X.
3. CONCLUSION
In general toplogy g-closed sets has a major role. Since its
inception several weak forms of g-closed sets have been
introduced in general toplogy. The present paper investigated
in new weak form of g closed sets namely gd-closed sets and
functions namely gd-irresolute functions in the light of d-open
sets in topological spaces. Some of its basic properties and
composition of functions is also discussed. Many examples
had been given to justify the results.
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International Journal of Computer Applications (0975 – 8887)
Volume 89 – No 18, March 2014
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